Small zeros of quadratic forms outside of a union of varieties

TL;DR

This paper proves the existence of small-height zeros of quadratic forms outside a union of varieties over global fields, extending classical results and providing explicit bounds.

Contribution

It establishes new bounds for small zeros of quadratic forms outside varieties, generalizing Cassels' theorem with explicit height estimates.

Findings

Existence of small-height zeros outside varieties

Construction of small-height maximal totally isotropic subspaces

Explicit height bounds independent of the variety’s height

Abstract

Let $F$ be a quadratic form in $N \geq 2$ variables defined on a vector space $V \subseteq K^N$ over a global field $K$, and $\Z \subseteq K^N$ be a finite union of varieties defined by families of homogeneous polynomials over $K$. We show that if $V \setminus \Z$ contains a nontrivial zero of $F$, then there exists a linearly independent collection of small-height zeros of $F$ in $V\setminus \Z$, where the height bound does not depend on the height of $\Z$, only on the degrees of its defining polynomials. As a corollary of this result, we show that there exists a small-height maximal totally isotropic subspace $W$ of the quadratic space $(V,F)$ such that $W$ is not contained in $\Z$. Our investigation extends previous results on small zeros of quadratic forms, including Cassels' theorem and its various generalizations. The paper also contains an appendix with two variations of Siegel's lemma. All bounds on height are explicit.